978-0128012420 Chapter 6

Solution Manual Chapter 6: Aeronautical

Engineering

1. Complete the following table where

1i

.

i-3

i-2

i-1

i0

i1

i2

i3

i4

i-5

i6

i

-1

–i

1

i

−1

−i

1

i

−1

2. Expand (1 + i)2 = ? (Answer: 2i)

3. Carry out the multiplication (3 + 2i)(1 + 7i) = ? (Answer: −11 + 23i)

4. What is (a + bi)(a − bi) = ? (Answer: a2 + b2)

5. The absolute value of a complex number is defined to be the distance from the origin

to the number in the Argand plane using the Pythagorean Theorem. What is the

absolute value of z = x + iy? (Answer: the absolute value of z is

22

xy

)

22

xy

i



4

11

27 i

i



7

i



7

i



7

50

149



7. Simplify:

 

i

31

23 2





.

 

   

10

27i31 

















i

ii

i

31

125

31

4129

31

23 2

2

8. What are the real and imaginary parts of A = 1/z2? Write your answer in the form

Write x2 – y2 as F to simplify the algebra:

 

      

     

 



 







































































22

2

2222

2

22

22222222222

4

2

4

2

44

2

4

2

22

2

yxyx

xyi

yxyx

yx

Z

yxF

xyi

yxF

F

yxF

xyiF

xyiFxyiF

xyiF

9. Look up Euler’s/de Moivre’s Theorem in your high school mathematics text book (or

the Interenet). It says that ei



= cos + isin where  is the rotational angle in the complex

plane1. Draw lines corresponding to  = 0, /4, /2, 3/4 and . What are their x, y

values on the surface of a unit circle?

1 You can see that the translation between polar coordinates and Cartesian coordinates is easy using

standard trig functions, e.g., x = R cos = cos and y = R sin = sin if R = 1 etc.

150. m/s if the air density is 0.525 kg/m3 and the circulation is 55.0 m2/s?

Need: Total lift on a 21.0 m span airfoil.

Know: The airfoil is travelling at 150. m/s, in air at a density of 0.525 kg/m3.

Circulation is 55.0 m2/s. Acceleration due to gravity is 9.81 m/s2.

11. In Example 6.8, the SFC for a Boeing 787 is 0.587 lbm/hr/lbf leading to a fuel

consumption rate of 12,680 lbm/hr. If a standard aviation jet fuel (JP– 4) costs $3.50/US

gallon, has a density of 6.84 lbm/US gallon, how much does fuel cost for each hour of

level flight?

Need: Fuel costs for a wide bodied jet.

Know: SFC is 0.587 lbm/hr/lbf leading to a fuel consumption rate of 12,680

lbm/hr. Fuel density is 6.84 lbm/US gallon. One US gallon costs $3.50.

12. An aircraft has a specific fuel consumption SFC of 0.450 lbm/hr per lbf of thrust.

Assume this is a constant during normal steady cruise conditions. The thrust over this

period is a constant 5,120 lbf. On a flight lasting 2.00 hours how much fuel is consumed

by this aircraft? How much is the cost of fuel over this 2 hour flight?

Assume 1.00 lbm of fuel = 0.150 US gallons and that it costs $3.50/US gallon. Ignore the

weight changes due to fuel consumption en-route.

Need: Fuel consumed in a 2.00 hr flight.

Know: SFC = 0.450 lbm/hr/lbf thrust. Fuel costs $3.50 per US gallon.

1.00 lbm of this fuel = 0.150 US gallons; thrust = 5,120 lbf.

13. Most large passenger planes fly at a cruising speed of about 0.85 × Mach 12. The

Mach number is defined as the ratio of actual speed V to the speed of sound, c.

The Mach number is an important variable in most speeding objects moving at or

exceeding the local speed of sound. The speed of sound is temperature sensitive as the √T

with T in kelvin. At 20ºC the speed of sound is 343 m/s or 767 mph.

At an altitude of 10,000 m, the atmosphere has cooled to -45ºC. What is the speed of

sound? (m/s and mph please).

Need: Speed of sound in air at 20ºC and at -45ºC in m/s and mph.

Know: Mach number varies as square root of absolute temperature. At 293K,

Mach 1 = 343 m/s or 767 mph.

2 Ernst Mach was an Austrian physics known especially for his work on shock waves. You have heard a

shock wave when lightning strikes nearby.

14. Refer to Example 6.8: How much extra fuel does it take if another passenger

(including luggage) is added to the plane for a total mass gain of 175 lbm? Assume the

flight time is 5.25 hours and the SFC is constant during level cruising at 0.587

(lbm/hr)/(lbf of thrust). The original thrust was 21,600 lbf.

Need: Extra fuel required for an extra passenger of mass 175 lbm.

Know: Original condition of plane:

Mass of plane + passengers = 324,000 lbm,

Thrust = 21,600 lbf;

Total lift to drag = 15.0

SFC was and remains at 0.587 (lbm/hr)/(lbf of thrust).

Fuel consumption = 12,680 lbm/hr.

Flight time was 5.25 hours so the total fuel consumed was 66,570 lbm.

3 Exclusive of the cost to the airline of a meal of peanuts!

15. A new development in airfoil design is the so called “scimitar winglet” as seen below

on a United Airlines airplane.

Actually this particular picture shows a double scimitar winglet, one facing upward and

one facing downward. Boeing states these winglets save several percent in drag (and thus

fuel) by reducing the size of vortices shed from the wingtip in normal flight. On a Boeing

737 these winglets stand as much as 3.5 m over the wingtip surfaces.

What’s the net improvement in lift-over-drag ratio given the unmodified plane had a lift–

to–drag ratio of 17.0 while the drag has been reduced by 4.00%?

Your solution must also compensate for the combined mass of 220.0 lbm of two winglets.

Assume the unmodified weight was 176,000 lbf.

Need: By adding scimitars to an existing aircraft, calculate the

improvement in its L/D ratio over a standard wing foil.

Know: When plane is in steady level flight L/D = 17.0 and lift = weight =

176,000 lbf. Also numerically lbm = lbf.

Kosky, Balmer, Keat and Wise: Exploring Engineering, Fourth Edition

16. Using the results of Exercise 15 what is the reduction in fuel rate of consumption in

steady level flight? Assume the fuel rate is proportional to the thrust when in steady level

flight.

Need: How much fuel is offset by scimitar winglets assuming the fuel use

is proportional to the thrust?

17.7.

17. Nevil Shute was a British/Australian author and an aeronautical engineer and highly

successful in both careers. He wrote an influential book entitled “No Highway” about a

post-World War II British airliner he called the “Reindeer”4.

The narrator is a hard driving government employed aeronautical engineer, Dr. Scott,

who takes over a laboratory staffed by over-the-hill specialists, most of whom he quickly

retires. But one metallurgist, Mr. Honey, is a quiet and unassuming scientist (but iron

willed). He is working on stress fracture5 of the tail plane of the Reindeer. Mr. Honey

says the tail unit will catastrophically fail in a certain number of flight hours and Dr.

Scott wonders if Mr. Honey is just another over-the-hill specialist. In fact a full scale

experiment of the Reindeer tail section of Mr. Honey’s is already underway to test for

stress fractures. Mr. Honey’s projection for a catastrophic failure is already overdue.

Then a Reindeer crashes killing all on board in remote Canada (while unfortunately

carrying a Soviet ambassador and causing an international incident).

Could Mr. Honey have been right all along? Should Dr. Scott have grounded the

Reindeer when he first heard of Mr. Honey’s theory? Or was his impression of Mr.

Honey as ineffectual correct?

Is there an ethical dilemma here? If Dr. Scott thinks Mr. Honey is wrong why has Mr.

Honey been allowed to waste the government’s money on a large and useless

experiment? On the other hand, if Dr. Scott believes Mr. Honey, why didn’t he ground

the Reindeer immediately?

Use an Engineering Ethics Matrix with these top line choices for Dr. Scott (you can add

more of your own).

4 The world’s first passenger jet was the Comet. (Reindeer and Comet seem to be related by a well-known

Santa poem!) The Comet did suffer from stress fatigue cracking and several aircraft went down with 100%

fatalities by the time the problem was fixed.

5 Take a piece of a tin can and bend it back and fro. It will eventually crack. This is a stress fracture.

Kosky, Balmer, Keat and Wise: Exploring Engineering, Fourth Edition

Actions for

Dr.Scott

Canons

Fire Mr.

Honey.

Encourage

Mr. Honey’s

expt.

Ground all

Reindeers

until

fracture

possibility

ruled out

Effects on

British

aeronautical

engineering

Report to a

newspaper

reporter

1) Hold

paramount the

safety, health

and welfare of

the public.

No. Mr. Honey

might be right

The only

way for this

is to keep the

expt. running

Yes.

“Paramount”

allows no

discussion.

Ground the

Reindeers.

–

2) Perform

services only in

the area of your

competence

Yes. This is

well within Dr.

Scott’s

purview

Yes, within

his

competence

Yes

Not really

qualified to

discuss

Yes, you

might for

the

government

3) Issue public

statements only

in an objective

and truthful

manner

Yes. Within

Dr. Scott’s

responsibilities

Yes, within

Dr. Scott’s

competence

Yes

Not qualified

Yes, you

might

through the

government

4) Act for each

employer or

client as

faithful agents

or trustees

Yes, but you

are acting for

the

government,

not the airline

nor the aircraft

manufacturer

Unclear if

encouraging

Mr. Honey

or stopping

him is the

correct path

Not clear

which

action to

take.

–

Yes, acting

for the

government.

5) Avoid

deceptive acts

This is only

deceptive if

you “ground”

Mr. Honey and

let the

Reindeer fly.

Not

deceptive

Not

deceptive

Not deceptive

Not

deceptive if

through the

government

6) Conduct

themselves

honorably …

Saving lives is

honorable but

what do you

owe the

airlines and

your

government?

Has the

potential for

saving lives.

If on best

available

data,

either

course is

honorable.

If and only if

Reindeer is

grounded and

Mr. Honey

continues

expt.

Yes you

might

through the

government